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''discrete valuation ring, discrete valuation domain''
 
''discrete valuation ring, discrete valuation domain''
  
A ring with a discrete [[Valuation|valuation]], i.e. an integral domain with a unit element in which there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331801.png" /> such that any non-zero ideal is generated by some power of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331802.png" />; such an element is called a uniformizing parameter, and is defined up to multiplication by an invertible element. Each non-zero element of a discretely-normed ring can be uniquely written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331803.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331804.png" /> is an invertible element and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331805.png" /> is an integer. Examples of discretely-normed rings include the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331806.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331807.png" />-adic integers, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331808.png" /> of formal power series in one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d0331809.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318010.png" />, and the ring of Witt vectors (cf. [[Witt vector|Witt vector]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318011.png" /> for a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318012.png" />.
+
A ring with a discrete [[Valuation|valuation]], i.e. an integral domain with a unit element in which there exists an element $  \pi $
 +
such that any non-zero ideal is generated by some power of the element $  \pi $;  
 +
such an element is called a uniformizing parameter, and is defined up to multiplication by an invertible element. Each non-zero element of a discretely-normed ring can be uniquely written in the form $  u \pi  ^ {n} $,  
 +
where $  u $
 +
is an invertible element and $  n \geq  0 $
 +
is an integer. Examples of discretely-normed rings include the ring $  \mathbf Z _ {p} $
 +
of $  p $-
 +
adic integers, the ring $  k [[ T ]] $
 +
of formal power series in one variable $  T $
 +
over a field $  k $,  
 +
and the ring of Witt vectors (cf. [[Witt vector|Witt vector]]) $  W ( k) $
 +
for a perfect field $  k $.
  
A discretely-normed ring may also be defined as a local principal ideal ring; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with group of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318013.png" />.
+
A discretely-normed ring may also be defined as a local principal ideal ring; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with group of values $  \mathbf Z $.
  
The completion (in the topology of a local ring) of a discretely-normed ring is also a discretely-normed ring. A discretely-normed ring is compact if and only if it is complete and its residue field is finite; any such ring is either isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318015.png" /> is a finite field, or else is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318016.png" />.
+
The completion (in the topology of a local ring) of a discretely-normed ring is also a discretely-normed ring. A discretely-normed ring is compact if and only if it is complete and its residue field is finite; any such ring is either isomorphic to $  k [[ T ]] $,  
 +
where $  k $
 +
is a finite field, or else is a finite extension of $  \mathbf Z _ {p} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318017.png" /> is a local homomorphism of discretely-normed rings with uniformizing elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318021.png" /> is an invertible element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318022.png" />. The integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318023.png" /> is the ramification index of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318024.png" />, and
+
If $  A \subset  B $
 +
is a local homomorphism of discretely-normed rings with uniformizing elements $  \pi $
 +
and $  \Pi $,  
 +
then $  \pi = u \Pi  ^ {e} $,  
 +
where $  u $
 +
is an invertible element in $  B $.  
 +
The integer $  e = e ( B / A ) $
 +
is the ramification index of the extension $  A \subset  B $,  
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318025.png" /></td> </tr></table>
+
$$
 +
[ B / \Pi B : A / \pi A ]  = f ( B / A )
 +
$$
  
is called the residue degree. This situation arises when one considers the integral closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318026.png" /> of a discretely-normed ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318027.png" /> with a field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318028.png" /> in a finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318030.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318031.png" /> is a semi-local principal ideal ring; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318032.png" /> are its maximal ideals, then the localizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318033.png" /> are discretely-normed rings. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318034.png" /> is a separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318035.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318036.png" />, the formula
+
is called the residue degree. This situation arises when one considers the integral closure $  B $
 +
of a discretely-normed ring $  A $
 +
with a field of fractions $  K $
 +
in a finite extension $  L $
 +
of $  K $.  
 +
In such a case $  B $
 +
is a semi-local principal ideal ring; if $  \mathfrak n _ {1} \dots \mathfrak n _ {s} $
 +
are its maximal ideals, then the localizations $  B _ {i} = B _ {\mathfrak n _ {i}  } $
 +
are discretely-normed rings. If $  L $
 +
is a separable extension of $  K $
 +
of degree $  n $,  
 +
the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318037.png" /></td> </tr></table>
+
$$
 +
\sum _ {i = 1 } ^ { s }  e ( B _ {i} / A ) f ( B _ {i} / A )  =  n
 +
$$
  
is valid. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318038.png" /> is a Galois extension, then all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318039.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318040.png" /> are equal, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318042.png" /> is a complete discretely-normed ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318043.png" /> itself will be a discretely-normed ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318044.png" />. On these assumptions the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318045.png" /> (and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318046.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318047.png" />) is known as an unramified extension if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318048.png" /> and the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318049.png" /> is separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318050.png" />; it is weakly ramified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318051.png" /> is relatively prime with the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318052.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318053.png" /> is separable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318054.png" />; it is totally ramified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318055.png" />.
+
is valid. If $  L / K $
 +
is a Galois extension, then all $  e ( B _ {i} / A ) $
 +
and all $  f ( B _ {i} / A ) $
 +
are equal, and $  n = sef $.  
 +
If $  A $
 +
is a complete discretely-normed ring, $  B $
 +
itself will be a discretely-normed ring and $  e ( B / A ) f ( B / A ) = n $.  
 +
On these assumptions the extension $  A \subset  B $(
 +
and also $  L $
 +
over $  K $)  
 +
is known as an unramified extension if $  e ( B / A ) = 1 $
 +
and the field $  B / \mathfrak n $
 +
is separable over $  A / \mathfrak m $;  
 +
it is weakly ramified if $  e ( B / A ) $
 +
is relatively prime with the characteristic of the field $  A / \mathfrak m $
 +
while $  B / \mathfrak n $
 +
is separable over $  A / \mathfrak m $;  
 +
it is totally ramified if $  f ( B / A ) = 1 $.
  
 
The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [[#References|[3]]]. Any module of finite type is a direct sum of cyclic modules; a torsion-free module is a flat module; any projective module or submodule of a free module is free. However, the direct product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of modules of rank one.
 
The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [[#References|[3]]]. Any module of finite type is a direct sum of cyclic modules; a torsion-free module is a flat module; any projective module or submodule of a free module is free. However, the direct product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of modules of rank one.
Line 21: Line 86:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Kaplansky,  "Modules over Dedekind rings and valuation rings"  ''Trans. Amer. Math. Soc.'' , '''72'''  (1952)  pp. 327–340</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Kaplansky,  "Modules over Dedekind rings and valuation rings"  ''Trans. Amer. Math. Soc.'' , '''72'''  (1952)  pp. 327–340</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318056.png" /> be a discretely-normed ring with uniformizing parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318057.png" />. The associated [[Valuation|valuation]] is then defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318058.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318060.png" /> a unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318061.png" />. A corresponding norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318062.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318065.png" /> is a real number between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318067.png" />. This makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318068.png" /> a normal ring. If the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318070.png" /> is finite it is customary to take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318071.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318072.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033180/d03318073.png" />.
+
Let $  A $
 +
be a discretely-normed ring with uniformizing parameter $  \pi $.  
 +
The associated [[Valuation|valuation]] is then defined by $  \nu ( a) = n $
 +
if $  a = u \pi  ^ {n} $,  
 +
$  u $
 +
a unit of $  A $.  
 +
A corresponding norm on $  A $
 +
is defined by $  | a | = c ^ {\nu ( a ) } $,
 +
$  | 0 | = 0 $,  
 +
where $  c $
 +
is a real number between 0 $
 +
and $  1 $.  
 +
This makes $  A $
 +
a normal ring. If the residue field $  k = A ( \pi ) $
 +
of $  A $
 +
is finite it is customary to take $  c = q  ^ {-} 1 $
 +
where $  q $
 +
is the number of elements of $  k $.

Latest revision as of 19:36, 5 June 2020


discrete valuation ring, discrete valuation domain

A ring with a discrete valuation, i.e. an integral domain with a unit element in which there exists an element $ \pi $ such that any non-zero ideal is generated by some power of the element $ \pi $; such an element is called a uniformizing parameter, and is defined up to multiplication by an invertible element. Each non-zero element of a discretely-normed ring can be uniquely written in the form $ u \pi ^ {n} $, where $ u $ is an invertible element and $ n \geq 0 $ is an integer. Examples of discretely-normed rings include the ring $ \mathbf Z _ {p} $ of $ p $- adic integers, the ring $ k [[ T ]] $ of formal power series in one variable $ T $ over a field $ k $, and the ring of Witt vectors (cf. Witt vector) $ W ( k) $ for a perfect field $ k $.

A discretely-normed ring may also be defined as a local principal ideal ring; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with group of values $ \mathbf Z $.

The completion (in the topology of a local ring) of a discretely-normed ring is also a discretely-normed ring. A discretely-normed ring is compact if and only if it is complete and its residue field is finite; any such ring is either isomorphic to $ k [[ T ]] $, where $ k $ is a finite field, or else is a finite extension of $ \mathbf Z _ {p} $.

If $ A \subset B $ is a local homomorphism of discretely-normed rings with uniformizing elements $ \pi $ and $ \Pi $, then $ \pi = u \Pi ^ {e} $, where $ u $ is an invertible element in $ B $. The integer $ e = e ( B / A ) $ is the ramification index of the extension $ A \subset B $, and

$$ [ B / \Pi B : A / \pi A ] = f ( B / A ) $$

is called the residue degree. This situation arises when one considers the integral closure $ B $ of a discretely-normed ring $ A $ with a field of fractions $ K $ in a finite extension $ L $ of $ K $. In such a case $ B $ is a semi-local principal ideal ring; if $ \mathfrak n _ {1} \dots \mathfrak n _ {s} $ are its maximal ideals, then the localizations $ B _ {i} = B _ {\mathfrak n _ {i} } $ are discretely-normed rings. If $ L $ is a separable extension of $ K $ of degree $ n $, the formula

$$ \sum _ {i = 1 } ^ { s } e ( B _ {i} / A ) f ( B _ {i} / A ) = n $$

is valid. If $ L / K $ is a Galois extension, then all $ e ( B _ {i} / A ) $ and all $ f ( B _ {i} / A ) $ are equal, and $ n = sef $. If $ A $ is a complete discretely-normed ring, $ B $ itself will be a discretely-normed ring and $ e ( B / A ) f ( B / A ) = n $. On these assumptions the extension $ A \subset B $( and also $ L $ over $ K $) is known as an unramified extension if $ e ( B / A ) = 1 $ and the field $ B / \mathfrak n $ is separable over $ A / \mathfrak m $; it is weakly ramified if $ e ( B / A ) $ is relatively prime with the characteristic of the field $ A / \mathfrak m $ while $ B / \mathfrak n $ is separable over $ A / \mathfrak m $; it is totally ramified if $ f ( B / A ) = 1 $.

The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [3]. Any module of finite type is a direct sum of cyclic modules; a torsion-free module is a flat module; any projective module or submodule of a free module is free. However, the direct product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of modules of rank one.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967)
[3] J. Kaplansky, "Modules over Dedekind rings and valuation rings" Trans. Amer. Math. Soc. , 72 (1952) pp. 327–340

Comments

Let $ A $ be a discretely-normed ring with uniformizing parameter $ \pi $. The associated valuation is then defined by $ \nu ( a) = n $ if $ a = u \pi ^ {n} $, $ u $ a unit of $ A $. A corresponding norm on $ A $ is defined by $ | a | = c ^ {\nu ( a ) } $, $ | 0 | = 0 $, where $ c $ is a real number between $ 0 $ and $ 1 $. This makes $ A $ a normal ring. If the residue field $ k = A ( \pi ) $ of $ A $ is finite it is customary to take $ c = q ^ {-} 1 $ where $ q $ is the number of elements of $ k $.

How to Cite This Entry:
Discretely-normed ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discretely-normed_ring&oldid=46738
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article